gistfile1.txt

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   "source": [
    "> It is useful to have a single measure of exposure for each risk factor whether the risk is   dichotomous, polytomous or continuous. This allows examination of trends in risk factors over time. An age-standardized version allows for comparison between places that highlights differences in exposure"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Our general form for this measure for a risk-outcome pair"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ SE_{joasct} = \\frac{\\int_{x=l}^{\\mu}RR_{joas}(x)P_{jasct}(x)dx - 1}{RR_{joas}(max) - 1} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> We average the values across different outcomes. Because the relative shape of the relative   risk curves across outcomes are generally consistent, the variation in the summary exposure  measure across outcomes for a risk is relatively small ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generate a single summary measure of exposure"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ SE_{jasct} = \\frac{\\sum_{o=1}^wSE_{joasct}}{w} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Preparing data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $RR_{max}$ (relative risk at the maximum level of exposure)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Source: /home/j/temp/stan/sev_draw/(exp_2015_06_18 | exp_2015_08_03)/"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### RR (relative risk)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Source: Mollie's getting data function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### P (exposure)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Source: Mollie's getting data function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Parameters "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> * inv_exp: flag indicating whether risk is beneficial; 1 indicates beneficial, 0 indicates detrimental\n",
    "* rr_scalar: scalar to standardize units of relative risk and exposure\n",
    "* integ_min: lower level of integration\n",
    "* integ_max: upper level of integration\n",
    "* tmrel_min: minimum TMREL \n",
    "* tmrel_max: maximum TMREL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Source: /home/j/temp/stan/GBD_2015/risks/risk_variables.xlsx (ask Stan if he updates them, he is still working on development)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calculating"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Getting parameters $( \\mu(p), \\sigma(p), RR_{mean} )$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "get_params(df_exp,df_rr,acause,age_group_id,location_id,year,sex)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Getting TMREL (Theoretical Minimum Risk Exposure Level)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "get_tmrel(risk,age,sex,iso3,tmrel_min,tmrel_max)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> If risk is among diet_pufa, metab_bmd, nutrition_iron, TMREL data is stored under /snfs3/WORK/05_risk/02_models/02_results; otherwise, it the mean of minimum TMREL and maximum TMREL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating exposure "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> If the distribution of exposure conforms to log normal distribution:\n",
    "\n",
    "> Function: /homes/jiaweihe/fbd/sev/cont_sev/draws_cont_sev/testlib.so\n",
    "\n",
    "> Relevant formula: https://en.wikipedia.org/wiki/Log-normal_distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\mu = ln(\\frac{\\mu(p)}{\\sqrt{1+\\frac{\\sigma^2(p)}{\\mu^2(P)}}}) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\sigma = \\sqrt{ln(1 + \\frac{\\sigma^2(p)}{\\mu^2(p)})}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$P(x) = \\frac{1}{x\\sigma\\sqrt{2\\pi}}e^{-{\\frac{(lnx - \\mu)^2}{2\\sigma^2}}} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> If the distribution of exposure conforms to normal distribution:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$P(x) = \\frac{1}{\\sigma(p)\\sqrt{2\\pi}}e^{-\\frac{(x - \\mu^2(p)}{2\\sigma^2(p)}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating relative risk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> If the risk factor is beneficial, the relative risk (RR) can be calculated by:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ RR(x) = RR_{mean}^{TMREL - x} $$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> If the risk factor is detrimental, the relative risk (RR) can be calculated by:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ RR(x) = RR_{mean}^{x - TMREL} $$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> If calculate relative risk is greater than maximum relative risk, then set relative risk as maximum relative risk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$RR(x) = RR_{max}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> If exposure level is greater than TMREL, then set relative risk as 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$RR(x) = 1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Getting SEV for a risk-outcome pair"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ SE_{joasct} = \\frac{\\int_{x=l}^{\\mu}RR_{joas}(x)P_{jasct}(x)dx - 1}{RR_{joas}(max) - 1} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Getting SEV for a risk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ SE_{jasct} = \\frac{\\sum_{o=1}^wSE_{joasct}}{w} $$"
   ]
  }
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